Tractor beams for colloidal particles are not self-accelerating modes*

We previously have demonstrated that structured laser beam can act as a tractor beam for colloidal particles. In addition to their ability to pull objects upstream, these modes of light have other remarkable properties. Due to the homology of the paraxial wave equation with Schrödinger’s equation, the wave function for a quantum mechanical particle in a circular box can be prepared in shape-preserving wave packets that rotate at constant angular speed around the center of the box, similar to a tractor beam. This apparent violation of Ehrenfest’s theorem is resolved by considering the force exerted on the particle’s wave packet by the enclosing wall. Remarkably, the wave packet continues to rotate even after the wall potential is removed. We show that this force-free finite-energy rotating state actually corresponds to classical motion with constant velocity, again in agreement with Ehrenfest’s theorem. Even so, the classical angular momentum carried by the rotating states poses a conceptual challenge because it differs from its quantum mechanical angular momentum, and indeed can have the opposite sign.